# Tagged Questions

In general topology, dimension theory studies various notions of dimension defined for topological spaces, for example Lebesgue covering dimension, small and large inductive dimension or Hausdorff dimension. In commutative algebra, dimension can be defined for commutative rings.

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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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### How to find family of functions

I want to find the family of functions with this property: $$\epsilon F\left(\frac{x}{\epsilon}\right)=\epsilon^r F(x)\quad\text{with}\ x,\epsilon, r \in \mathbb{R}$$ ($\epsilon$ and r are constants)...
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### Infimum of fractal dimensions left by almost-disjoint-disk covering of plane

Answers to this question show that it is not possible to cover the plane by almost-disjoint closed disks. As note in the comments, the Appollonian Gasket leaves a set of fractal dimension ...
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### Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same center). ...
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### Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of closed ...
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### Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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### Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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### Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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### Hochschild dimension

I'm curious; if $A$ ia a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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### Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
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### Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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### How can i evaluate a parabolic region in 2-Dimensions onto n-Dimensions?

I am working on the K-Nearest Neighbor algorithm and have the solution to a problem as a parabola in 2-dimensional space. Should I continue this solution onwards to n-dimensions (and thereby create ...
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### Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
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### Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated $\... 0answers 45 views ### Covering dimension of the union Let$\{{A_n}\}$be the closed subsets of$X$, such that${A_n} \subset \operatorname{Int}{A_{n + 1}}$and$ \cup {A_n} = X$, if$A_1$and all$\operatorname{cl}(A_n-A_{n-1})$have the covering ... 0answers 58 views ### Disjointness of stars in a simplicial complex in$\ell_2$Definitions Let's consider a full$n$-dimensional simplicial complex$C$in$\ell_2(X)$. By that I mean a set of all functions$f:X \to[0,1]$such that$\sum_{x\in X}f(x)=1$and there are at most$n+...
I have been trying to (rigorously) get the Minkowski's dimension (refer here for a basic definition) of a parabola: $(x,y) \in \mathbb{R}^2 : \{ y = x^2\}$.
Let $F$ be a closed subset of a manifold $M$ of dimension $n$ such that $M\setminus F$ is disconnected and F contains no open sets. Is that true the small inductive dimension of F is $n-1$? Small ...